Let $f : V \to U$ be a (generalized) polynomial-like map. Suppose that
harmonic measure $\omega= \omega(\cdot,\infty)$ on the Julia set $J_{f}$ is equal to
the measure of maximal entropy $m$ for $f : J_{f} \hookleftarrow$. Then the
dynamics $(f,V,U)$ is called maximal. We are going to give a criterion for the
dynamics to be conformally equivalent to a maximal one, that is to be
conformally maximal. In the second part of this paper we construct an
invariant ‘harmonic’ measure $\mu$ such that ${d\mu}/{d\omega}$ is Hölder
for certain dynamics. This allows us to prove in this class of dynamical
systems that $\omega\approx m$ is necessary and sufficient for $(f,V,U)$ to be
conformally maximal. In the particular case when $f$ is expanding and $J_{f}$
is a circle, our result becomes a theorem of Shub and Sullivan; so throughout
the paper we are dealing with an analog of a theorem of Shub and Sullivan on
‘wild’ (e.g. totally disconnected) $J_{f}$ and for certain non-expanding
$f$. We also construct (under certain assumptions) invariant harmonic measure on
$J_{f}$. In this respect, our work stems from one of the works of Carleson.